Question: Solve for $x$ : $ 5|x - 9| + 4 = -1|x - 9| + 10 $
Solution: Add $ {1|x - 9|} $ to both sides: $ \begin{eqnarray} 5|x - 9| + 4 &=& -1|x - 9| + 10 \\ \\ { + 1|x - 9|} && { + 1|x - 9|} \\ \\ 6|x - 9| + 4 &=& 10 \end{eqnarray} $ Subtract ${4}$ from both sides: $ \begin{eqnarray} 6|x - 9| + 4 &=& 10 \\ \\ { - 4} &=& { - 4} \\ \\ 6|x - 9| &=& 6 \end{eqnarray} $ Divide both sides by ${6}$ $ \dfrac{6|x - 9|} {{6}} = \dfrac{6} {{6}} $ Simplify: $ |x - 9| = 1$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x - 9 = -1 $ or $ x - 9 = 1 $ Solve for the solution where $x - 9$ is negative: $ x - 9 = -1 $ Add ${9}$ to both sides: $ \begin{eqnarray} x - 9 &=& -1 \\ \\ {+ 9} && {+ 9} \\ \\ x &=& -1 + 9 \end{eqnarray} $ $ x = 8 $ Then calculate the solution where $x - 9$ is positive: $ x - 9 = 1 $ Add ${9}$ to both sides: $ \begin{eqnarray} x - 9 &=& 1 \\ \\ {+ 9} && {+ 9} \\ \\ x &=& 1 + 9 \end{eqnarray} $ $ x = 10 $ Thus, the correct answer is $x = 8 $ or $x = 10 $.